The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
a'#(proper(x)) |
→ |
a'#(ok(x)) |
(32) |
a'#(f(f(active(x)))) |
→ |
a'#(f(g(f(mark(x))))) |
(23) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[a'#(x1)] |
= |
1 · x1
|
together with the usable
rules
mark(f(X)) |
→ |
f(mark(X)) |
(15) |
mark(top(X)) |
→ |
proper(top(X)) |
(21) |
f(active(X)) |
→ |
active(f(X)) |
(14) |
f(proper(X)) |
→ |
proper(f(X)) |
(16) |
g(proper(X)) |
→ |
proper(g(X)) |
(18) |
ok(f(X)) |
→ |
f(ok(X)) |
(19) |
ok(g(X)) |
→ |
g(ok(X)) |
(20) |
ok(top(X)) |
→ |
active(top(X)) |
(22) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[a'#(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 1 · x1
|
[active(x1)] |
= |
1 |
[g(x1)] |
= |
1 |
[mark(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 + 1 · x1
|
together with the usable
rules
ok(f(X)) |
→ |
f(ok(X)) |
(19) |
ok(g(X)) |
→ |
g(ok(X)) |
(20) |
ok(top(X)) |
→ |
active(top(X)) |
(22) |
f(active(X)) |
→ |
active(f(X)) |
(14) |
f(proper(X)) |
→ |
proper(f(X)) |
(16) |
g(proper(X)) |
→ |
proper(g(X)) |
(18) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
a'#(f(f(active(x)))) |
→ |
a'#(f(g(f(mark(x))))) |
(23) |
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[a'#(x1)] |
= |
1 · x1
|
together with the usable
rules
ok(f(X)) |
→ |
f(ok(X)) |
(19) |
ok(g(X)) |
→ |
g(ok(X)) |
(20) |
ok(top(X)) |
→ |
active(top(X)) |
(22) |
g(proper(X)) |
→ |
proper(g(X)) |
(18) |
f(active(X)) |
→ |
active(f(X)) |
(14) |
f(proper(X)) |
→ |
proper(f(X)) |
(16) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
2 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 + 3 · x1
|
[top(x1)] |
= |
3 + 2 · x1
|
[active(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 + 2 · x1
|
[a'#(x1)] |
= |
3 · x1
|
the
pair
a'#(proper(x)) |
→ |
a'#(ok(x)) |
(32) |
and
the
rules
ok(g(X)) |
→ |
g(ok(X)) |
(20) |
ok(top(X)) |
→ |
active(top(X)) |
(22) |
g(proper(X)) |
→ |
proper(g(X)) |
(18) |
could be deleted.
1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
mark#(f(X)) |
→ |
mark#(X) |
(30) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(f(X)) |
→ |
mark#(X) |
(30) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
ok#(g(X)) |
→ |
ok#(X) |
(38) |
ok#(f(X)) |
→ |
ok#(X) |
(36) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[g(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[ok#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
ok#(g(X)) |
→ |
ok#(X) |
(38) |
|
1 |
> |
1 |
ok#(f(X)) |
→ |
ok#(X) |
(36) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
f#(proper(X)) |
→ |
f#(X) |
(31) |
f#(active(X)) |
→ |
f#(X) |
(28) |
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[f#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(proper(X)) |
→ |
f#(X) |
(31) |
|
1 |
> |
1 |
f#(active(X)) |
→ |
f#(X) |
(28) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
g#(proper(X)) |
→ |
g#(X) |
(34) |
1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[g#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(proper(X)) |
→ |
g#(X) |
(34) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.